Universal Gravitation

Lecturer: Professor Stephen T. Thornton

Reading Quiz

A) Equal to her weight on Earth.

B) A little less than her weight on Earth.

C) Less than half her weight on Earth.

D) Zero (she is weightless).

E) Somewhat larger than her weight on Earth.

The International Space Station is at an altitude of 200 km above the surface of the Earth. What is the net force on an astronaut at rest inside the Space Station?

Answer: B

The astronaut is falling around the Earth. The gravitational force is keeping her from going in a straight line. The acceleration of gravity is a little less than g at 200 km above the Earth.

200 km

6380 km

Last Time

Non-uniform circular motion

Drag

Terminal velocity

Fundamental forces

Today

History of gravitation

Newton’s law of universal gravitation

Kepler’s laws

History of GravitationGreeks used the geocentric frame in which the Earth was at the center.

Ptolemy, 2nd A.D., prepared a detailed formulation of heavenly body motion.

epicycles

retrograde motion

Nicolaus Copernicus – (1473-1543 ) introduced heliocentric frame with the Sun at the center of the solar system.

Catholic church thought this was heresy.

Danish astronomer Tycho Brahe (1546- 1601) made huge number of observations. Johannes Kepler continued his work and did analysis.

Galileo (1564-1642) is said to have invented the telescope. He made many observations.

Issac Newton (1642-1727) was one of the smartest persons to ever live.

He invented calculus so he could solve the problem of how the moon rotates around Earth.

Newton looked at Kepler’s results and figured everything out in a manner of days!

Newton’s Law of Universal Gravitation

If the force of gravity is being exerted on objects on Earth, what is the origin of that force?

Newton’s realization was that the force must come from the Earth.

He further realized that this force must be what keeps the Moon in its orbit.

conic sections

Newton showed that planetary motion and other similar motion had to be of conic sections.

Newton’s Law of Universal Gravitation

The force of gravity between any two point objects is attractive and of magnitude:

1 22

m mF G

r

G is the universal gravitational constant11 2 26.67 10 N m / kgG

Gravitational Force Between Point Masses

1 22

m mF G

r

Important points about gravity

1) Gravitational force is a vector and always attractive.

2) Gravity is difficult to measure, except for large bodies.

3) Gravity has an exact 1/r2 dependence.

4) For several masses, just add forces. Called superposition.

Gravitational Force Between a Point Mass and a Sphere (uniform mass density)

Use symmetry. For this case, force acts at center.

MmF

mMF

Gravitational Force Between the Earth and the Moon

EmF

mEF

EmF

mEF

Conceptual QuizA) the Earth pulls harder on the MoonA) the Earth pulls harder on the MoonB) the Moon pulls harder on the EarthB) the Moon pulls harder on the EarthC) they pull on each other equallyC) they pull on each other equallyD) there is no force between the Earth D) there is no force between the Earth

and the Moon and the MoonE) it depends upon where the Moon is E) it depends upon where the Moon is

in its orbit at that timein its orbit at that time

Which is stronger, Earth’s pull on the Moon, or the Moon’s pull on Earth?

By Newton’s 3rd Law, the forces are equal and opposite.

Conceptual QuizA) the Earth pulls harder on the MoonA) the Earth pulls harder on the MoonB) the Moon pulls harder on the EarthB) the Moon pulls harder on the EarthC) they pull on each other equallyC) they pull on each other equallyD) there is no force between the Earth D) there is no force between the Earth

and the Moonand the MoonE) it depends upon where the Moon is E) it depends upon where the Moon is

in its orbit at that timein its orbit at that time

Which is stronger, Earth’s pull on the Moon, or the Moon’s pull on Earth?

Conceptual QuizA) one quarterA) one quarterB) one halfB) one halfC) the sameC) the sameD) two timesD) two timesE) four timesE) four times

If the distance to the If the distance to the Moon were doubled, Moon were doubled, then the force of then the force of attraction between attraction between Earth and the Moon Earth and the Moon would be:would be:

The gravitational force depends inversely on the distance squared. So if you increaseincrease the distancedistance by a factor of 22, the forceforce will decreasedecrease by a factor of 44.

Conceptual QuizA) one quarterA) one quarterB) one halfB) one halfC) the sameC) the sameD) two timesD) two timesE) four timesE) four times

If the distance to the If the distance to the Moon were doubled, Moon were doubled, then the force of then the force of attraction between attraction between Earth and the Moon Earth and the Moon would be:would be:

2RMm

GF

Follow-up:Follow-up: What distance would What distance would increaseincrease the force by a factor of the force by a factor of 22??

Force of gravity on a mass m on the surface of the Earth is mg.Let’s use Newton’s universal law.

2

2

2

11 24

22 6 2

224

N m

kgkg

m

(6.67 10 )(5.98 10 )9.80 m/s

(6.38 10 )

Assume we know , but not . Solve for :

5.98 10 kg

E

E

E

E

E E

EE

mMF mg G

R

GMg

R

g M M

gRM

G

Kepler’s three laws follow naturally from Newton’s Law of Universal Gravitation.

First law occurs because of 1/r2. Orbits must be ellipses.

We will derive Kepler’s laws later after we study angular momentum.

Kepler’s 1st and 2nd laws

2nd law: Radius vector sweeps out equal areas in same time interval.

http://physics.bu.edu/~duffy/semester1/semester1.html

Kepler’s 3rd law follows directly from the form of the gravitational force law.

22 3

3/2

4

2

S

S

T rGM

T rGM

Kepler’s Third Law and Some Near Misses

Free FallLook at

http://galileoandeinstein.physics.virginia.edu/more_stuff/Applets/newt/newtmtn.html

Gravitational Attraction. Two objects attract each other gravitationally with a force of when they are 0.25 m apart. Their total mass is 4.00 kg. Find their individual masses.

102.5 10 N

Sun’s Mass Determination. Determine the mass of the Sun using the known value for the period of the Earth and its distance from the Sun. [Hint: The force on the Earth due to the Sun is related to the centripetal acceleration of the Earth.] Compare your answer to that given in your textbook.

Conceptual Quiz

The gravitational constant G is  A) equal to g at the surface of Earth. B) different on the Moon than on Earth. C) obtained by measuring the speed of falling objects having different masses. D) all of the above. E) none of the above

Answer: E

None of them determine G.

Newton’s Law of Universal Gravitation

Using calculus, we can show:

Particle outside a thin spherical shell: gravitational force is the same as if all mass were at center of shell.

Particle inside a thin spherical shell: gravitational force is zero. See next slide.

Can model a sphere as a series of thin shells; outside any spherically symmetric mass, gravitational force acts as though all mass is at center of sphere.